Calculated using standard mathematical and industry formulas.
Free online Triangle Calculator calculator for academic, engineering, and statistical math.
How to use this calculator?
Select the combination of known measurements: three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA), or two angles and a non-included side (AAS).
Enter the known values in the fields shown.
All remaining sides, angles, area, perimeter, and altitudes appear instantly.
For right triangles, use the dedicated Right Triangle Calculator for additional outputs.
Formula & calculation method
Law of Cosines & Heron's Formula
c² = a² + b² − 2ab·cos(C) · Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
Law of Cosines applies for SSS/SAS. Law of Sines (a/sin A = b/sin B = c/sin C) applies for ASA/AAS. Heron's formula requires all three sides.
Step-by-step calculation example
Triangle with sides 3, 4, 5: s = 6, Area = √(6·3·2·1) = √36 = 6. Angles: A = 36.87°, B = 53.13°, C = 90°.
Key insights to remember
For a 3-4-5 right triangle, the area is 6 square units and the hypotenuse is exactly 5 — the simplest Pythagorean triple.
The Law of Cosines generalizes the Pythagorean theorem: c² = a² + b² − 2ab·cos(C).
The triangle inequality requires every side to be strictly less than the sum of the other two.
An equilateral triangle with side s has area (√3/4)s² and all three 60° angles.
The inradius r = Area / s where s is the semi-perimeter; the circumradius R = abc / (4·Area).
How to interpret your results?
Area (Heron's formula)
Computed from the semi-perimeter s and all three sides: Area = √(s(s−a)(s−b)(s−c)). Works for any triangle once all sides are known.
Triangle type
Classified by sides (scalene, isosceles, equilateral) and by largest angle (acute, right, obtuse). This affects how area and height distribute across the triangle.
Height (altitude)
The altitude to side a is hₐ = 2·Area / a. In architecture and surveying, altitudes are used to measure effective perpendicular distances.
Frequently asked questions
How does the Triangle Calculator calculator work?
It uses standard algebraic, geometric, or statistical theorems to return exact values.
How accurate are the results?
All calculations use double-precision floating-point arithmetic for exact accuracy.
Can I copy the step-by-step work?
Yes, step-by-step solutions are formatted clearly for homework or documentation.
How are undefined operations handled?
The calculator flags mathematical errors such as division by zero or negative square roots.
Is this tool suitable for students and engineers?
Yes, it is designed for students, teachers, engineers, and scientists.
Does it support negative numbers and decimals?
Yes, all real numbers (positive, negative, fractions, and decimals) are supported.
Related calculators
Right Triangle Calculator — Solve a right triangle from any two known values — legs, hypotenuse, or an angle — using the Pythagorean theorem.
Area Calculator — Calculate the area and perimeter of rectangles, squares, circles, triangles, trapezoids, ellipses and parallelograms.